Complex numbers question bank final - Kar
Complex numbers
One mark questions
1. Express the following in x+iy form
i)
ii)
iii)
iv)
v) vi) (1+2i)(2-3i)
vii)
viii)find x and y given that x+iy=
ix)If 3 4
?
x)Find the multiplicative inverse of 2-3i
xi)Find the multiplicative inverse of
xii) If z= Prove that ? 1 2. Find the Modulus and Amplitude of the following
iii)
3. Find 4. Simplify the following
i) cos
cos isin cos isin
Two mark questions:
1. If x+iy=
then Prove that x y
2. Find the Modulus and Amplitude of the following
i)1+iii) 1 3
iii 3 iv) 33 3 v) vi)
vii)1+cos+isin
viii) 1+cos
ix)1+
x)
xi)
3. Prove that
cos isin
4.
icot
5.
=itan
6. If z
are complex numbers, then prove the
following each carries 2 marks
i |z ? z |=|z | ? |z | ii)Amp z ? z =Amp z Amp z
iii
||
= iv)Amp
= Amp z
||
Amp z
7. Simplify the following
i) sin
ii)
iii)
iv) 1 cos isin V) 1 cos
8. Find the value of 1 cos 9. Find the value of
1 cos isin
1 cos isin
10.Prove that
1
11.If x+ 2cos and If y+ 2cos , then prove the
following(each carries 2 marks)
i)
=2cos( ) ii)
=2isin( )
iii)
=2cos(m
)
iv) xy =2cos
12. If , is an imaginary cube root of unity then show tha the following(each carries 2 marks)
i) (1+ - (1+ =0 ii) (1+
iii) (1+
iv) (1
= -8
v) (1+ vi) (1-
(1+
1
)4 = 16
vii) (1+ (1+ ) (1+ ) (1+ ) = 1
viii) (1+ 2 ix) 1 x) 1
1 = 64 1
2 = 16
2 27 0
xi) 1
+ 1
= 32
xii)
0
xiii) (3- ) (3- (3-
3
163
xiv) (1+ - 3 ? (1+ - 3 =0
xv) (1+ - (1+ - (1-
(1-
16
Four mark questions:
1. If z=cos+isin, then Prove that
2. If z=cos+isin, then Prove that
3. Prove the following
i) 1 3i 8
ii) 1 3i
1 3i 16
itann itan2
iii 1 3i 1 3i 128
iv 3 i 3 i 0
v) 3 i 3 i 0 323
Vi) 1 3i
1 3i
2
3. If 2-2x+4=0 thenProve that + =27
4. If cos+cos 0 and sin+sin 0, then Prove that
i) cos2+cos2 2cos ii) sin2+sin2 2sin
5. Find the cube roots of the following and represent them in Argand's diagram. Also find their continued product.
i)1+i ii) 8i iii) 43 4 iv) 3
v) 3
6. Find the fourth roots of the following and represent them in Argand's diagram. Also find their continued product.
i) 3
ii) 1 3 i iii)3 ? i
7. Find all the values of
8. Find all the values of z= 3
, and also
find their continued product and represent them
in Argand'sdiagram
9. Solve the equation z3=3 i 10. Solve x7-x4+x3-1=0 11. solve x5 ? x3 + x2 - 1=0. And represent cube
roots of unity in Argand diagram
Six mark questions:
1. State and Prove Demoivre's theorem
2. If cos
0
then
Prove the following
i) Cos3
3
ii) Sin3 +sin3
(iii) cos2
(iv) Sin2
3 3cos 33
3/2 3/2
v)cos( )+ cos( )+ cos( )=0
vi) sin( )+ sin
)+ sin( )=0
................
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